Walking through a meadow we encounter two kinds of flowers, daisies and dandelions. As we walk in a straight line, we model the positions of the flowers we encounter with a one-dimensional Poisson...

Walking through a meadow we encounter two kinds of flowers, daisies and dandelions. As we walk in a straight line, we model the positions of the flowers we encounter with a one-dimensional Poisson process with intensity λ. It appears that about one in every four flowers is a daisy. Forgetting about the dandelions, what does the process of the daisies look like? This question will be answered with the following steps.

a. Let Nt be the total number of flowers, Xt the number of daisies, and Yt be the number of dandelions we encounter during the first t minutes of our walk. Note that Xt + Yt = Nt. Suppose that each flower is a daisy with probability 1/4, independent of the other flowers. Argue that

Since it is clear that the numbers of daisies that we encounter in disjoint time intervals are independent, we may conclude from c that the process (Xt) is again a Poisson process, with intensity λ/4. One often says that the process (Xt) is obtained by thinning the process (Nt). In our example this corresponds to picking all the dandelions.